Equation of state of nuclear matterand nuclei in laboratories and in neutron-star crusts
Kazuhiro Oyamatsu (Aichi Shukutoku U.), Kei Iida (RBRC)LARGEST UNCERTAINTY
L :densty-derivative coefficient ofsymmetry energy200
3L (MeV)150

y=-350 [MeV fm ]C3
y=-220 [MeV fm ]TM1100

F

B
50

A
DE0G

H

I
y=-1800
SIII-50200250
300
K0 (MeV)350

Macroscopic properties of nuclei in labs. and in neutron stars＝＞ We adopt a macroscopic model.
1. Masses and radii os stable nuclei => We construct about 200 EOS's systematically.
2. Focus on saturation parameters of nearly symmetric nuclear matterand identify the allowed region of these EOS parameter values.
3. Calculate masses and radii of unstable nuclei in labs. => dominant EOS parameter
4. Calculate nuclei in neutron-star crusts. => dominant EOS parameter
proton number, average proton fraction of spherical nuclei
pasta-nucleus phase :lower and upper boundary density

Energy per nucleon of nearly symmetric nuclear matter
22K0ªw n, x w0 ++ 1 – 2 x S0 + L n – n02 n – n0
3n 0
18n 0
n0 :nuclear density, w0 :saturation density, K0 :incompressibility
S0 :symmetry energy at n=n0, L:its density derivative coefficientspherical nuclei and pasta nuclei

uniform
matterK.Oyamatsu, NPA561, 431 (1993)

What dominates the existence of pasta nuclei in crusts?
○ Yes! There are pasta nuclei.
Compressible liquid-drop modelFPS interaction (fitted to FP EOS)
Lorenz, Ravenhall and Pethick, PRL70(1993)
Uncertainties in m p and surface tension
Watanabe, Iida and Sato, NPA676(2000)
Thomas-Fermi modeloya1-4 (fitted to masses and sizes of stable nuclei and FP EOS)
K.Oyamatsu, NPA561(1993)
DBHF EOS
Sumiyoshi, Oyamatsu and Toki, NPA595(1995)
× No! No pasta nuclei can exist.
Compressible liquid-drop model (SKM interaction)
SKM interaction (Lorenz, Ravenhall and Pethick, PRL70(1993))
SLy4, SLy7 (Douchin, Haensel and Meyer, NPA665(2000))
Relativistic Thomas-Fermi model (Euler-Lagrange Eq.)
RMF Lagrangian (Cheng, Yao and Dai, PRC55(1997))

The subnuclear density EOS dominates the existence.
=>Let's study empirically allowed EOS systematically!

Step 1
Generate all empirically allowed EOS'ssystematicallyAdopted macroscopic model
Energy per cell
dr e 0 n n, n p + m nn n + m pn p +

W=2

dr F0

n n + electron kinetic energy + Coulomb

cellcellnn (np) : local neutron (proton) density,n=nn+np : total density

e 0(nn, np) : EOS of uniform nuclear matter (energy density)
F0 : surafce energy parameter
Parametrization of the EOS (energy density)e= 3 3p 2n,np0 n52/322h n 5/3 + h n 5/3 + 1 2m n n2m p p

Fermi kinetic energy density

1 - 2Yp

2v s(n) + 1 - 2Yp

2v n(n)potential energy density

potential energy densities of symmetric and neutron matter
33anbn2vs(n) = a 1n 2+ 2vn(n) = b 1n 2 +1 + a 3n
1 + b 3n★a1〜b2 and F0：masses and radii of stable nuclei (b3=1.59 fm3, a fit to FP EOS）
★very flexible function form
a3 can vary K0 widely. (better than Skyrme）
The function can be fitted to SIII and TM1 EOS very well.

Simplified Thomas-Fermi calculationenergy minimization with respect to parameters of nn(r) and np(r) (and lattice constant)
0.10

1–n ini –n outi
ni r =90

0.08

-3

r ti
Ri

3density (fm )neutron (proton) density distribution nn (np)
+ n out
in outir < Rir < RiRn (Rp) : neutron (proton) radius parametertn (tp) : neutron (proton) surface thickness parameterniin : central desnity
nnout : neutron gas density (npout=0)

Zr0.06

exp.
calc.

0.04

stable nucleus

0.02

charge distribution

0.00
024r (fm)68

A good function form
The n and p distributions are independent.
=> neutron skin
The empirical information is limited:radius and thickness.
The gradient term in Euler Eq. is continuous.
The density is zero beyond the classical turning point.

The values of parameters a1〜b3(EOS) and F0 are determined
to fit masses and radii of stable nuclei.
=> about 200 sets of empirical EOS+F0L
EOS parameter values obtained
from stable nuclei

KS0:symmetry energyL :density symmetry coefficient

9 representative EOS

A-IA

BCDEFGHI

Step 2
Calculations with the 200 EOS's
Neutron-rich unstable nuclei in labs

The mass, radius and neutron skin are dependent on L.

Step 3 (Calculations with 200 EOS's)
Nuclei in neutron star crusts
Density region of pasta nuclei

Z and Yp are small for large L.
For large L, S(n) is small at n<n0.

Estimate of density region of pasta nuclei
lower boundary
stability against fission of spherical nuclei
In the liquid drop model, (Coulomb self energy)=2*(surface energy)==> (volume fraction of nucleus) = 1/8

upper boundary (core-crust boundary)
instability against forming proton clusters>0v(Q)≈The upper bound (core-crust boundary density) is clearly dependent
on L while the lower is almost constant.

Summary
１．The values of L and K0 cannot be determined from masses
and radii of stable nuclei.
２．Radii and masses of unstable nuclei have appreciable
sensitivity to L.
３．The core-crust boundary density is dependent on L.
４．The existence of the pasta phase is dominated by L.
The pasta phase exists if L < 100 MeV.
５．The present uncertainty in L is too large.